Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-7x+y &= -1 \\ -9x+2y &= -1\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}14x-2y &= 2\\ -9x+2y &= -1\end{align*}$ Add the top and bottom equations. $5x = 1$ Divide both sides by $5$ and reduce as necessary. $x = \dfrac{1}{5}$ Substitute $\dfrac{1}{5}$ for $x$ in the top equation. $-7( \dfrac{1}{5})+y = -1$ $-\dfrac{7}{5}+y = -1$ $y = \dfrac{2}{5}$ $y = \dfrac{2}{5}$ The solution is $\enspace x = \dfrac{1}{5}, \enspace y = \dfrac{2}{5}$.